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Theorem fmptss 28279
Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. (Contributed by Thierry Arnoux, 30-May-2020.)
Assertion
Ref Expression
fmptss  |-  ( A 
C_  B  ->  (
x  e.  A  |->  C )  C_  ( x  e.  B  |->  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem fmptss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3458 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 566 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  =  C
)  ->  ( x  e.  B  /\  y  =  C ) ) )
32ssopab2dv 4749 . 2  |-  ( A 
C_  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  C_  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) } )
4 df-mpt 4484 . 2  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
5 df-mpt 4484 . 2  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
63, 4, 53sstr4g 3505 1  |-  ( A 
C_  B  ->  (
x  e.  A  |->  C )  C_  ( x  e.  B  |->  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   {copab 4481    |-> cmpt 4482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-in 3443  df-ss 3450  df-opab 4483  df-mpt 4484
This theorem is referenced by:  carsgclctunlem2  29159
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